Kazimierz Brudzewski*
Use of the Mueller Matrices in Ellipsometry
The Mueller technique is adequate to describe most optical systems using polarized light. The use of the Mueller matrices for ellipsometric calculation is very promising. Application of this method to analysis of the ellipsometer arrangement suggests that a new arrangement might be constructed. This new type of the optical system can be constructed without rotating elements and modulators. The new type of the ellipsometer works on the principle of the independent and simultaneous measurements of all the Stokes parameters [1],
Introduction
Ellipsometry is a very sensitive and versatile optical method for analysis of specularly-reflecting surfaces or of films deposited on them. The changes of the polarization state of the light reflected from a surface are measured by means of that technique. Such measurements enable to calculate the optical constants of a surface as well as the thickness and refractive index of a thin film overlaying that surface. The technique, instrumentation, and calculations have previously been described [2], [5], [6], However, so far**), these calculations have been carried out by use of the classical method, without application of the matrices. In this paper application of the Mueller matrices to ellipsometry is presented. A mathematical model of the ellipsometer arrangement is obtained by the Mueller technique. In this model the polarized light is represented by a Stokes vector and each component of the optical system is described by 4 x 4 matrices called the Mueller matrices. The Mueller matrix for the whole optical system can be found by multiplication of matrices for the indi vidual parts.
The analysis o f the ellipsometric arrangement
The fundamental optical arrangement of the ellipso meter is shown in Fig. 1. From Fig. 1 it is seen that the Mueller matrix for a whole optical system can be found by multiplication of the matrices
1 A S Q -cos 2P sin 2 P 0 ( 1)
* Institute of Physics Technical University, Warsaw. ** Mac Crackin applied the matrieces calculations analysis of instrumental errors in ellipsometry [3],
where 1 cos2 P sin 2 P
0
— the Stokes vector of the light from the polarizer.
Q, S, A — Mueller matrices of the compen
sator, surface and analyser
P — azimuth of the polarizer (see
Fig. 2)
Fig. 1. Optical components in an ellipsometer: P — polarizer, Q — compensator, S — surface of the sample, A — analyser
Fig. 2. The azimuth reference frame. The azimuth of the plane of transmitted polarization of polarizer and analyser is P and A. The azimuth of the fast axis of the compensator is a. The axis x of the right-handed Cartesian coordinate system
is parallel to the plane of incidence
The Mueller matrix of the ideal compensator (ideal compensator introduces only the phase shift) is given by Q ( y , a ) = R{ - 2a)B(y )R{2a) (2) Α = Ψ and P = - Δ 2 1 o = - Ψ and P = - Δ Y +45° (8)
where R and B operators describe rotations about the center of v the Poincare sphere,
y denotes a relative phase retardation
along axes of the compensator,
a denotes an azimuth of the fast axis
of the compensator. The matrix for the surface is
Ξ(Ψ, Δ) = (3)
1 — cos 2Ψ 0 0
— cos2¥/ 1 0 0
0 0 sin2V/ coszl sin2¥'sinzl 0 0 -sin2'i/sinzl sin2¥'coszl where Ψ and I are two ellipsometric parameters defined from the known relations. Multiplying the Stokes vector of the light from the polarizer by the Mueller matrices of the compensator and the surface gives the vector of the light entering the analyser.
1 SCR, A )R (-2 a)B (y )R (2 a) cos2 P
sin2P 0
(4)
If the compensator is a quarter-wave plate with the azimuth of the fast axis 45° then
a = 45° y = 90°
Two sets of solutions, called zones, are obtained. In an analogous way one may derive expressions for the case of an imperfect compensator. The imperfect compensator (imperfect quarter-wave plate) is cha racterized by the ratio T, of the transmittance along its slow axis to the transmittance along the fast axis and the relative phase retardation y. The Mueller matrix of the imperfect quarter-wave plate can be written
Q(y, a) = R ( —2a)D(y, T)R(2a) (9) Assuming that T is near 1 and y near 90° then we can use only the first-order approximation for the matrix D 1 ε 0 0 ε 1 0 0 D( y, T) * 0 0 - δ 1 0 0 - 1 - δ T = 1+ ε y = 90°+<5
Substituting (9) and (10) into (4) we have the vector of the light entering the analyser
l+£sin2F+<5cos2¥/cos2/> —cos2¥/ (l + £ sin 2 />) —<5cos2P sin2¥/ cosZl (ε + sin2F )+ sin2l// sin Δ cos2 P —sin2i/ sinZl (£ + sin 2 P )+ sin2!Fcoszl cos2P
. (11)
Substituting this value into expression (4), we have From (11), we find two equations 1
—cos2¥/ sin2¥rsin(2.P+ Δ) sin2¥/cos(2P+ Δ)
Since this light is extinguished by the analyser, it must be linearly polarized, therefore
sin2¥'cos(2/>+ J ) = 0 (6) and the azimuth A of the analyser is
tg2A = —tg2¥/ sin(2/>+ z l) (7) The solution equation (6) and (7) can be obtained in the form
(5)
—sin2'// sinZl (ε + sin2P )+ sin2¥/ coszl cos2P
= 0 (12)
and
sin 2y>cos Δ (ε + sin2P) + sin2w sin Δ cos 2 P
tg2A = ---
---—cos2^)(l + £ sin 2 />) — <5 cos 2 P (13) Similarly we obtain two solutions of equations (12) and (13): and A = w-\---- <5sinzlsin 2 Δ o 1 P --- 45°----2 2 2ψ - csinZl, (14)
A = -wA---- ό sin zl sin2w
Ψ 2 ‘
Sur I’utilisation de la matrice de mueller en ellipsometrie
and
Δ 0 1
P = --- f45 T esinzl.
2 2
The terms with the second and higher powers of ε and <5 in (14) are neglected. It should be emphasized that these two solutions are in good agreement with results obtained previously [3], [5],
Dans ce travail on a introduit une notation operationnelle pour la matrice de Mueller qui facilite l’analyse des systemes ellipsometriques. On a presente un modele mathematique des systemes ellipsometriques a 1’aide de la methode de Mueller. On a etabli aussi l’expression pour des defauts de l’ellipso- metre dus au fait que le compensateur n ’est pas parfait. Enfin, on a propose un nouveau type du systeme ellipsometrique.
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From relation (3) it can be easely seen that the ellipsometrie parameters Δ and ψ can be obtained as the ratio of the Stokes parameters. For example, if a linearly polarized light beam with azimuth +45° falls on the sample, then the ratio of the Stokes parameters gives Δ and ψ in the form
5 M
tgZl = — — coslip = --- — (15) where I, M, C, S — Stokes parameters (we use the
notation of Shurcliff [4]). Therefore the measurements of the Stokes parameters are a hopeful experimental method in ellipsometry. This is of particular value for the automatic ellipso- meter construction.
The author is most grateful to doc. dr Halina Ch^cmska for her help and interest in the work.
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References
[1] Brudzewski K., Grochowski L., Patent P-144843 (1970). [2] Mac Crackin F. L., Passagalia E., Stromberg R. R.,
Steinberg H. L., ResJ., Natl. Bur. Std.. 67A 363 (1963). [3] Mac Crackin F. L., JOSA, Vol. 60, No 1, 57 (1970). [4] ShurcliffW. A., BallardS. S., Polarized Light, Warsaw
PWN 1968.
[5] SmithP. H., Surface Science, Vol. 16, 34 (1969).
[6] Winterbottom A. B., Trans. Faraday Soc. 42, 487 (1946).
